\def\ptl{\partial}
\def\uv{{\bf u}}
\def\Uv{{\bf U}}
\def\Av{{\bf A}}
\def\Rv{{\bf R}}
\def\vv{{\bf v}}
\def\wv{{\bf w}}
\def\nv{{\bf n}}
\def\del{\nabla}
\nopagenumbers
Fluid Dynamics I \hfil PROBLEM SET 7 \hfil Due November 3, 2003
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1. Consider the Joukowski airfoil with $\zeta_0=bi$ and $a>b > 0$. The circle in the $\zeta$-plane
passes through the points $(\pm a, 0)$. (a) Show that the airfoil is an arc of the circle with center at $(0, -(a^2-b^2)i/b$ and radius $(a^2+b^2)/b$. (b) With Kutta condition applied to the trailing edge, at what angle of attack (as a function of $a,b$)is the lift zero?
\vskip .1in
2. Consider a 3D wing of high aspect ratio. Let the airfoil parameters other than chord (i.e. $k,\beta $)
be independent of $y$,
the coordinate along the span of the wing.
Also, assume the planform is symmetric about the line $x = 0$ in the $x-y$
plane.
Using Prandtl's lifting-line
theory, show that for a given lift the minimal induced drag occurs for
a wing having an elliptical planform. Show in this case that the coefficient
of induced drag $C_{D_i}= 2\times drag / (\rho U^2 S)$ and lift
coefficient $C_L=2\times lift / (\rho U^2 S)$ are related by
$$C_{D_i} =C_L^2 /(\pi A).$$
Here $S$ is the wing area and $A$ is the aspect ratio $4b^2/S$. (Some of the WW II fighters,
notably the Spitfire, adopted an approximately elliptical wing.)
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3. A 3D body $D$ moves steadily with velocity $\Uv$. The flow is a potential
flow exterior of the body, $phi$ being the potential for the flow relative to stationary fluid at infinity,
and ${\partial \phi\over\partial n} = \Uv\cdot \nv$ on the body surface $\partial D$. Given that for large $R^2=x^2+y^2+z^2$ the potential decays like
$$\phi = -{a\over R}- {\Av\cdot\Rv\over R^3} + O(R^{-3}),$$
where $a,\Av$ are constants (scalar and vector respectively), show that necessarily $a=0$.
(Note: $\int_{\partial D} \nv\cdot\Uv dS = 0$.)
\vskip .1in
4. Justify the expression for total fluid energy exterior to the body in problem 2 above,
as measured relative to the stationary fluid at infinity:
$$E={1\over 2}\rho (4\pi \Av\cdot\uv-V_0 u^2),$$
where $V_0$ is the volume of the body.
\vskip .1in
5. From the results for 2D potential flow given in class, show that the apparent mass of a flat plate
of length 4a moving broadside on is $4\pi a^2\rho$.
\bye
\end